The problem is asking to simplify the following expression and give the answer as a single power of 2:

$\left(2^{\frac{3}{4}} \times 2^{\frac{5}{8}} \div 2^{\frac{1}{6}}\right)^4$

Step-by-step solution:

1. Simplify the exponents inside the parentheses: Using the laws of exponents: $a^m \times a^n = a^{m+n}, \quad \frac{a^m}{a^n} = a^{m-n}$ The expression inside the parentheses is: $2^{\frac{3}{4}} \times 2^{\frac{5}{8}} \div 2^{\frac{1}{6}} = 2^{\left(\frac{3}{4} + \frac{5}{8} - \frac{1}{6}\right)}$

2. Combine the exponents: To add and subtract these fractions, find the least common denominator (LCD). The denominators are 4, 8, and 6. The LCD of 4, 8, and 6 is 24. Convert each fraction to have a denominator of 24:

   $\frac{3}{4} = \frac{18}{24}, \quad \frac{5}{8} = \frac{15}{24}, \quad \frac{1}{6} = \frac{4}{24}$

   Now, combine the exponents: $\frac{18}{24} + \frac{15}{24} - \frac{4}{24} = \frac{29}{24}$

   So, the expression inside the parentheses simplifies to: $2^{\frac{29}{24}}$

3. Apply the outer exponent: Now raise the simplified expression to the power of 4: $\left(2^{\frac{29}{24}}\right)^4 = 2^{\frac{29}{24} \times 4} = 2^{\frac{116}{24}} = 2^{\frac{29}{6}}$

Thus, the final simplified expression is:

$2^{\frac{29}{6}}$

Let me know if you'd like more details or have any questions!

Here are 5 related questions you might find helpful:

1. How do you simplify expressions with exponents using fraction addition and subtraction?
2. What is the law of exponents for dividing numbers with the same base?
3. How do you convert fractions to have a common denominator?
4. How do you raise a number with an exponent to another power?
5. Can this type of simplification be applied to bases other than 2?

Tip: Always make sure to use the least common denominator when combining fractions with different denominators.